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PREFACE     .
decided that a certain theorem had really been proved? Might
not there be concealed inconsistencies in the very foundations
and postulate systems on which the whole elaborate structure
had been reared? It began to appear that an exhaustive re-
examination of everything from the ground up was demanded*
The capital problem was to prove the self-consistency of
mathematical analysis  the calculus and its numerous modern
offshoots. Presently this programme turned out to be far more
difficult than had been anticipated, and David Hilbert (1862-
1943), the last of the giants from the nineteenth century, in
1898 proposed the more modest problem of proving the consis-
tency of arithmetic. This led to the like for mathematical logic.
All was going well till 1931, when Kurt Godel (1906- )
showed that in any well-defined system of mathematical axioms
there exist mathematical questions which cannot be settled on
the basis of these axioms. But suppose we go to a more inclusive
system in which, perhaps, the questions can be settled. The
same difficulty appears in the new system, and so on indefi-
nitely. There are thus specific purely mathematical 'yes-no'
questions which will be forever undecidable by human beings.
This wholly unexpected conclusion has been called the most
significant advance in logic since Aristotle. It does not mean
that mathematics has gone to smash, but it does suggest that
some of the claims made for mathematics in the past will have
to be moderated. One philosophical die-hard who thoroughly
misunderstood what Godel had done, proudly proclaimed, *I
am an Aristotelian. The old logic is good enough for me', which
sounded like an echo of the revivalist hymn "The old-time
religion, the old-time religion is good enough for me.1 Aristo-
telian logic may be good enough for the old-timers, but it is
not good enough for mathematics, nor has it been for at least
three centuries. As one detail, Aristotle's logic makes no pro-
vision for variables and functions as they occur in mathematics*
There is not space here to elaborate any of this, but those
interested will find an elementary and lucid account by Alfred!
 Tarski in his Introduction to Logic and the Methodology of Deduc-
tive Sciences (Oxford University Press, 2nd Edition, 1941).
1953                                                            E . T.   B E Xi L